Pentagon Calculator

How to use the regular pentagon calculator

What this calculator does

This calculator solves the standard geometry of a regular convex pentagon from one known measurement. A regular pentagon has five equal sides and five equal interior angles. Once one size measurement is known, every other linear dimension is fixed by constant ratios, and the area is fixed by the square of the scale. The tool calculates Side, Perimeter, Area, Diagonal, Height, Circumcircle radius, and Incircle radius (apothem). It does not solve irregular pentagons, self-intersecting pentagrams, three-dimensional pentagonal prisms, or real objects whose sides and angles are only approximately regular.

When to use it

Use the calculator when checking a geometry exercise, sizing a regular pentagonal sign or panel, estimating material area from an edge or radius, or converting a drawing between metric and U.S. customary units. It is also useful for verifying that several stated dimensions describe the same regular pentagon. The area identity used here follows the standard regular-polygon relationship explained in Khan Academy's lesson on finding the area of a regular polygon from its side length.

How to calculate

1. The calculator opens with a ready-to-use demonstration: Side = 10 cm. Its results and a validated example workbook are available immediately.
2. Under Solve from, choose the one measurement you know. Selecting a different property while the current state is valid preserves the same pentagon and replaces the input with that property's equivalent value.
3. Enter the positive value. Use a decimal point, such as 12.5. Commas are accepted only in conventional groups, such as 1,250; decimal-comma input such as 1,5 is rejected to avoid ambiguity.
4. Choose the Unit. Length results use mm, cm, m, in, or ft; area uses the corresponding square unit. Changing the unit converts the active value and keeps the physical pentagon unchanged.
5. Read the primary Area, the six supporting result cards, and the Property details table. Then select Download Excel to export the current inputs and typed results as a real spreadsheet workbook.
6. Reset clears the demonstration value and all calculated content. Download Excel is then disabled until a complete valid measurement is entered again.

Input guide

Solve from is required and accepts one of seven named geometric properties. Choose Side for a measured edge, Perimeter for the total boundary, Area for enclosed surface, Diagonal for a vertex-to-nonadjacent-vertex segment, Height for the overall top-to-base distance in the standard orientation, Circumcircle radius for the center-to-vertex radius, or Incircle radius (apothem) for the center-to-side radius. A common mistake is treating a diagonal as a side or confusing the circumradius with the shorter apothem.

Side value, or the renamed value label for the selected property, is required. Enter a finite number greater than zero; an example is 10. Scientific notation such as 1e3 is accepted; mixed units, negative values, zero, and malformed separators are rejected. For Area, the number is interpreted in square units, so 10 cm² is not equivalent to a side of 10 cm. Larger linear inputs increase every linear output proportionally, while area increases with the square of the scale.

Unit is required and controls both input interpretation and output formatting. The available choices are millimeters, centimeters, meters, inches, and feet. The calculator uses exact factors for the international inch and foot. NIST documents both SI length units and why area must be expressed in squared units in its guide to SI units of area. The most frequent conversion error is converting an area with a linear factor instead of squaring that factor; this calculator performs the squared conversion automatically.

Output guide

Area is the exact regular-pentagon area implied by the input, displayed in square units. Side, Perimeter, Diagonal, Height, Circumcircle radius, and Incircle radius (apothem) are length outputs in the selected unit. All are mathematical identities for an ideal regular pentagon, subject only to display rounding. A zero result is not meaningful here because the input must be positive. Very large or very small valid results may appear in scientific notation to preserve legibility.

The Property details table repeats each output with its symbol and its relationship to side s. The Calculated value column is the current numeric result. The Relationship to side s column shows the formula used after the selected input has been converted into the canonical side length. The fixed header pills – five equal sides, 108° interior angle, 72° central angle, and the golden-ratio diagonal – describe every regular pentagon and do not change with size.

Worked example

With the startup example s = 10 cm, the perimeter is P = 5s = 50 cm. The area is A = s²√(25 + 10√5) / 4, which gives 172.0477 cm² after display rounding. The diagonal is d = s(1 + √5) / 2, giving 16.1803 cm. The same model produces a height of 15.3884 cm, a circumradius of 8.5065 cm, and an apothem of 6.8819 cm. These values match the first-open cards, table, and workbook checkpoints.